Of course I know how to do this for fixed quantities of $\mathbf{x}$ and $\mathbf{y}$. However, the corresponding probabilities do not allow for a covering argument. I am hoping that the upper bound for all $\mathbf{x},\mathbf{y}$ is comparable to the result for fixed $\mathbf{x},\mathbf{y}$ up to constant/log factors. If anybody knows of a counter argument which shows that this is not the case that would also be very helpful. Assume that $m \ge c n$ for a sufficiently large numerical constant $c$. I would also be ok with an argument which assumes $m \ge c n (\log n)^\alpha$ for some small $\alpha$ like $\alpha=1,2,3,4$ (the smaller of course the better).

Whether your hope is true or not, certainly depends on the relation between $m$ and $n$. Let's take two extremes: 1) $m=1$. Then the discrepancy factor is like $n^2$. 2) $m=\infty$. Then the law of large numbers tells you that your quantity is essentially the same for all $x$.
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fedjaMay 22 '14 at 17:57

Thanks! I forgot to mention the relationship between m and n. I have added this in the above
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mohiMay 22 '14 at 18:05